A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient

We consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient, such as

$?F(x,u,Du,D^{2}u)=\lambda c(x)u+〈M(x)Du,Du〉+h(x)$

in a bounded domain with a Dirichlet boundary condition; here $\lambda \in \mathbb{R}$, $c,h\in L^p(\mathrm{\Omega })$, $p>n\ge 1$, $c?0$ and the matrix M satisfies $0<{\mu }_{1}I\le M\le {\mu }_{2}I$. Recently this problem was studied in the “coercive” case $\lambda c\le 0$, where uniqueness of solutions can be expected; and it was conjectured that the solution set is more complex for noncoercive equations. This conjecture was verified in 2015 by Arcoya, de Coster, Jeanjean and Tanaka for equations in divergence form, by exploiting the integral formulation of the problem. Here we show that similar phenomena occur for general, even fully nonlinear, equations in nondivergence form. We use different techniques based on the maximum principle.We develop a new method to obtain the crucial uniform a priori bounds, which permit to us to use degree theory. This method is based on basic regularity estimates such as half-Harnack inequalities, and on a Vázquez type strong maximum principle for our kind of equations.     

Existence,nonexistence and multiplicity results for nonlocal Dirichlet problems

In this paper we establish various existence, nonexistence and multiplicity results for fully nonlinear Dirichlet problems associated to nonlocal Hamilton–Jacobi equations. This study is accomplished by a careful analysis of the principal eigenvalues of the elliptic operator. Resonance phenomena and anti maximum principles are also established.     

Free boundary problem for a fully nonlinear and degenerate parabolic equation in an angular domain

This paper concerns with the problem of how to running an insurance company to maximize his total discounted expected dividends. In our model, the dividend rate is limited in $[0,M]$ and the company is allowed to transfer any proportion of risk by reinsuring. So there are two strategies which we call dividend strategy and reinsurance strategy. The objective function and the corresponding optimal two strategies are the solution and the two free boundaries of the following Barenblatt parabolic equation

$v_t?\underset{0\le a\le 1}{\max }?(\frac{1}{2}{a}^2{\sigma }^2{v}_{xx}+a\mu v_x)+cv?\underset{0\le l\le M}{\max }?[(1?v_x)l]=0$

under certain boundary conditions on an angular domain

$Q_T=\{(x,t)|0<x<Mt,0<t\le T\}.$

The main effort is to analyze the properties of the solution and the free boundaries to show the optimal decision for the insurance company.     

关于不定方程x3-1=749y2

利用同余式、平方剩余、Pell方程的解的性质、递归序列证明了:不定方程x³-1=749y²仅有整数解(x,y)=(1,0).     

Stable blow-up dynamics in the -critical and -supercritical generalized Hartree equation

We study stable blow-up dynamics in the generalized Hartree equation with radial symmetry, which is a Schrödinger-type equation with a nonlocal, convolution-type nonlinearity: First, we consider the -critical case in dimensions and obtain that a generic blow-up has a self-similar structure and exhibits not only the square root blowup rate , but also the log-log correction (via asymptotic analysis and functional fitting), thus, behaving similarly to the stable blow-up regime in the -critical nonlinear Schrödinger equation. In this setting, we also study blow-up profiles and show that generic blow-up solutions converge to the rescaled , a ground state solution of the elliptic equation . We also consider the -supercritical case in dimensions . We derive the profile equation for the self-similar blow-up and establish the existence and local uniqueness of its solutions. As in the NLS -supercritical regime, the profile equation exhibits branches of nonoscillating, polynomially decaying (multi-bump) solutions. A numerical scheme of putting constraints into solving the corresponding ordinary differential equation is applied during the process of finding the multi-bump solutions. Direct numerical simulation of solutions to the generalized Hartree equation by the dynamic rescaling method indicates that the is the profile for the stable blow-up. In this supercritical case, we obtain the blow-up rate without any correction. This blow-up happens at the focusing level , and thus, numerically observable (unlike the -critical case). In summary, we find that the results are similar to the behavior of stable self-similar blowup solutions in the corresponding settings for the nonlinear Schrödinger equation. Consequently, one may expect that the form of the nonlinearity in the Schrödinger-type equations is not essential in the stable formation of singularities.     

Modified two-grid method for solving coupled Navier-Stokes/Darcy model based on Newton iteration

A new decoupled two-gird algorithm with the Newton iteration is proposed for solving the coupled Navier-Stokes/Darcy model which describes a fluid flow filtrating through porous media. Moreover the err...     

Self-affinity and nonextensivity of sunspots

In this paper we study the time series of sunspots by using two different approaches, analyzing its self-affine behavior and studying its distribution. The long-range correlation exponent α has been calculated via Detrended Fluctuation Analysis and the power law vanishes to values greater than 11 years. On the other hand, the distribution of the sunspots obeys a q-exponential decay that suggests a non-extensive behavior. This observed characteristic seems to take an alternative interpretation of the sunspots dynamics. The present findings suggest us to propose a dynamic model of sunspots formation based on a nonlinear Fokker–Planck equation. Therefore its dynamic process follows the generalized thermostatistical formalism.     

Travelling wave mathematical analysis and efficient numerical resolution for a one-dimensional model of solid propellant combustion

We investigate a model of solid propellant combustion involving surface pyrolysis coupled to finite activation energy gas-phase combustion. Existence and uniqueness of a travelling wave solution are established by extending dynamical system tools classically used for premixed flames, dealing with the additional difficulty arising from the surface regression and pyrolysis. An efficient shooting method allows to solve the problem in phase space without resorting to space discretisation nor fixed-point Newton iterations. The results are compared to solutions from a CFD code developed at ONERA, assessing the efficiency and potential of the method, and the impact of the modelling assumptions is evaluated through parametric studies.     

The stabilizability of heat equations with time-dependent coefficients

In this paper, we study the stabilizability of heat equations with time-dependent coefficients. The instability of the systems is illustrated by special solutions. The boundary feedback stabilization results are established by the backstepping method where the growth order of the kernel function depends on time.